Problem 1 - Constraining the cosmological parameters using the Planck power spectrum¶
Planck is the third-generation space telescope, following COBE and WMAP, and it aims to determine the geometry and content of the Universe by observing the cosmic microwave background radiation (CMB), emitted around 380,000 years after the Big Bang. Permeating the whole universe and containing information on the properties of the early Universe, the CMB is widely known as the strongest evidence for the Big Bang model.
Measuring the spectrum of the CMB, we confirm that it is very close to the radiation from an ideal blackbody, and flunctuations in the spectrum are very small. Averaging ocer all locations, its mean temperature is $2.725K$, and its root mean square temperature fluctuation is $\langle(\frac{\delta T}{T})^2\rangle^{1/2} = 1.1 \times 10^{-5}$ (i.e. the temperature of the CMB varies by only ~ 30 $\mu K$ across the sky).
Suppose you observe the fluctuations $\delta T/T$. Since we are taking measurements on the surface of a sphere, it is useful to expand $\delta T/T$ in spherical harmonics (because they form a complete set of orthogonal functions on the sphere):
$$ \frac{\delta T}{T} (\theta, \phi) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} \mathrm{a}_{lm} \mathrm{Y}_{lm} (\theta, \phi) $$
In flat space, we can do a Fourier transform of a function $f(x)$ as $\sum_k \mathrm{a}_k \mathrm{e}^{ikx}$ where $k$ is the wavenumber, and $|\mathrm{a}_k|$ determines the amplitude of the mode. For spherical harmonics, instead of $k$, we have $l$, the number of the modes along a meridian, and $m$, the number of modes along the equator. So $l$ and $m$ determine the wavelength ($\lambda = 2\pi/l$) and shape of the mode, respectively.
In cosmology, we are mostly interested in learning the statistical properties of this map and how different physical effects influence different physical scales, so it is useful to define the correlation function $C(\theta)$ and split the CMB map into different scales.
Suppose that we observe $\delta T/T$ at two different points on the sky. Relative to an observer, they are in direction $\hat{n}$ and $\hat{n}'$ and are separated by an angle $\theta$ given by $cos\theta = \hat{n} \cdot \hat{n}'$ Then, we can find the correlation function by multiplying together the values of $\delta T/T$ at the two points and average the product over all points separated by the angle $\theta$.
$$ C(\theta)^{TT} = \Big\langle \frac{\delta T}{T}(\hat{n})\frac{\delta T}{T}(\hat{n}') \Big\rangle_{\hat{n} \cdot \hat{n}' = cos\theta}$$
The above expression is specific to the temperature fluctuations, but we can also do a similar analysis for the polarization map of the CMB. (The CMB is polarized because it was scattered off of free electrons during decoupling.) We decompose the polarization pattern in the sky into a curl-free "E-mode" and grad-free "B-mode."
However, the CMB measurements (limited by the experiment resolution and the patch of sky examined) tell us about $C(\theta)$ over only a limited range of angular scales. (i.e. the precise values of $C(\theta)$ for all angles from $\theta = 0$ to $\theta = 180^\circ$ is not known.) Hence, using the expansion of $\delta T/T$ in spherical harmonics, we write the correlation function as:
$$ C(\theta) = \frac{1}{4\pi}\sum_{l=0}^\infty (2l+1) C_l P_l(cos\theta) $$
where $P_l$ are the Legendre polynomials.
So we break down the correlation function into its multipole moments $C_l$, which is the angular power spectrum of the CMB.
Remember that $\lambda = 2\pi/l$. So $C_l$ measures the amplitude as a function of wavelength. ($C_l = \frac{1}{2l+1}\sum_{m = -l}^l |\mathrm{a}_{lm}|^2$). In this problem, we will consider the temperature power spectrum $C_l^{TT}$, the E-mode power spectrum $C_l^{EE} = \frac{1}{2l+1}\sum_{m = -l}^l |\mathrm{a}_{lm}^E|^2$, and the temperature-polarization cross-correlation $C_l^{TE} = \frac{1}{2l+1}\sum_{m = -l}^l \mathrm{a}_{lm}^{T*} \mathrm{a}_{lm}^E$.
THe CMB angular power spectrum is usually expressed in terms of $D_l = l(l+1)C_l/2\pi$ (in unit of $\mu K^2$) because this better shows the contribution toward the variance of the temperature fluctuations.
Cosmologists built a software called "cosmological boltzmann code" which computes the theoretical power spectrum given cosmological parameters, such as the Hubble constant and the baryon density. Therefore, we can fit the theory power spectrum to the measured one in order to obtain the best-fit parameters.
Here, we consider six selected cosmological parameters, $\vec{\theta} = [\theta_1, \theta_2, ..., \theta_6] = [H_0, \Omega_b h^2, \Omega_c h^2, n_s, A_s, \tau]$. ($H_0$ = Hubble constant, $\Omega_b h^2$ = physical baryon density parameter, $\Omega_c h^2$ = physical cold dark matter density parameter, $n_s$ = scalar spectral index, $A_s$ = curvature fluctuation amplitude, $\tau$ = reionization optical depth.). We provide you with the measured CMB power spectra from Planck Data Release 2.
References :
http://carina.fcaglp.unlp.edu.ar/extragalactica/Bibliografia/Ryden_IntroCosmo.pdf (Chapter 9, Intro to Cosmology, Barbara Ryden)
